Put \(S\)  as the dimension of \(\mathfrak {u}\cap \mathfrak {k}\) . Cohomological induction functors (also called Zuckermann functors and Bernstein functors) lift \(Z\)  to \((\mathfrak {g}, K)\)  modules \(\mathcal {L}_j(Z)\)  and \(\mathcal {R}^j(Z)\) , where \(j\)  is a non-negative integer. We refer the reader to Chapter V of [1]} for details. Here we simply recall that the most interesting case happens at the middle degree \(S\) . For instance, if \(Z\)  is weakly good, then \(\mathcal {L}_j(Z)\)  and \(\mathcal {R}^j(Z)\)  vanish except for the degree \(S\) , while \(\mathcal {L}_S(Z)\cong \mathcal {R}^S(Z)\)  is zero or irreducible with infinitesimal character \(\Lambda _Z+\rho (\mathfrak {u})\) . Moreover, \(\mathcal {L}_S(Z)\)  is unitary provided \(Z\)  is unitary. When \(Z\)  is actually in the good range, we have that \(\mathcal {L}_S(Z)\)  must be non-zero, and it is unitary if and only if \(Z\)  is unitary.
